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  	\title {Fluid Structure Interaction : A Brief Review}
    \author{Chandrasekhar Jinendran} % \\ for new line 
    \date{\today}
 
 \begin{document}
	\maketitle 
 	
 	\section{Overview}
 	
 	Section \ref{sec:intro} shows an outline of the literature. Sections \ref{subsec:way} and \ref{subsec:approach} on page \pageref{subsec:way} covers the common methodologies in solving FSI problems. Section \ref{subsec:coupling} refers to the different methods of partitioned approach and section \ref{sec:disc} explains about the most common discretisation method in FSI named ALE(Arbitrary Lagrangian Eulerian) approach. Section \ref{sec:eqn} refers to the governing equations of FSI. Finally, section \ref{sec:bc} explains the boundary conditions imposed generally to an FSI problem.
 	
 	% ch: for chapter; fig: for figure; tbl: for table 
 	
 	\section{Introduction} 
 	\label{sec:intro}
 	A system is said to be coupled when two or more physical systems are interacting together and solution of each system is dependent on the solution of the other. An example of a coupled system is Fluid Structure Interaction where there is interaction between a fluid and a structure. Here neither the fluid part nor the structural part can be solved independently because of the unknown forces in the interface region. In fluid structure interaction, the solid and fluid systems exchange mechanical energy at the interface in both directions. The fluid exerts forces on the moving solid changing its dynamics and the structure causes displacements in the fluid changing the characteristics of flow. In many applications, the forces from the moving fluid deforms the structure only by small amount. In those cases the response from the structure can be solved independently when fluid flow characteristics are determined, since the deformation does not affect the flow. Systems needed to be coupled when the fluid forces causes considerable deformation of the structure causing change in the fluid pattern.
 	
 	Some of the complications arise from the fact that fluid and solid mesh systems are quite different. Frequently, the fluid system uses an Eulerian or spatially fixed-coordinate system, while the solid system uses a Lagrangian or material fixed-coordinate system. Hence, we must take care to develop a suitable interfacing technique between the two modules. Also, the time scales can be different for the two modules; hence one must be careful while performing coupled calculations.
 	
 	
 	\section {Theory} 
 	
 	\subsection{One-Way and Two-Way FSI}	
 	\label{subsec:way}
 	In One-Way FSI, either the fluid or structural domain is initially solved independently, and the results are employed as a model condition when solving the other domain. In a Two-Way FSI study, the fluid and structural domains are solved in parallel. At each sub-step the fluid and structural domain solutions are required to converge before moving to the next step\cite{bungartz}. The fluid and structural domains are connected through a coupling scheme, like the Arbitrary Langrangian Euler method, where the mesh of one domain is allowed to move so that it adheres to the boundaries of the other domain, or IB method, where the mesh does not deform and structural movement within the fluid is accounted for by adding body forces to the equations of motion. Fully-coupled Two-Way FSI studies are typically transient and capture fluid structure behavior over a period.

 	\subsection{Monolithic and Partitioned Approach}	
	\label{subsec:approach}
 	Generally FSI problems are too complex to solve analytically so their numerical simulations are very important. The two main approaches are monolithic and partitioned approach. In monolithic or direct approach, the governing equations of fluid and structural displacements are solved simultaneously. The discretisation of the problem results in a large number of equations which is solved with a single solver. This approach has the advantage of stability since the mutual influence of the solid and fluid can be taken into account directly. Whereas in partitioned approach, governing equations of the fluid and solid domains are solved separately with two separate solvers\cite{partition}. Here a coupling algorithm is required for interaction and for solving the coupled problem. The basic procedure of partitioned approach consists of exchange of data between fluid and structural solvers alternatively at the interface boundary for every instant.  It is easy to implement due to the freedom in choosing fluid and structural solvers.
 	
 	
 	Since a monolithic coupling procedure considers all the fluid and structure problem unknowns simultaneously at the same instant, the complete system becomes difficult to solve. So a partitioned coupling is preferred for real-world applications.
 	
 	\subsection{Loosely and Strongly Coupled}
 	\label{subsec:coupling}
 	In partitioned approach, interaction can be either loosely or strongly coupled. A loosely coupled algorithm is explicit and there will be bidirectional exchange of solved variables per time step sequentially. For loosely coupled algorithms stability is dependent on mass density ratio (ratio of solid density to fluid density) and geometry of the domains. Here stability decreases with decreasing mass density ratio. For unstable computations, decreasing of time step size will lead to earlier occurrence of instabilities or increased instabilities. For conditionally stable computations the time step size has an upper limit depending on the CFL number (Courant-Friedrichs-Lewy condition) and a lower limit depending on the highest eigen mode of the added mass matrix (since fluid closest to the coupling interface will act as extra mass on the structure).
 	
    Whereas a strongly coupled algorithm is implicit and performs iterations to converge at the end of time step. They are more stable for low mass density ratio than the loosely coupled algorithm.
    Decrease of this ratio leads to more sub-iterations which in turn leads to increased computing time.
    
 	
 	\section{Discretisation}
 	\label{sec:disc}
 	To account for the deformation of the mesh, ALE formulation of the Navier Stokes equations is used, discretized on a control volume with volume V(t) and boundary S(t). ALE is used in order to combine the advantages of the Lagrangian and Eulerian formulations, as they are usually exploited for individual structural and fluid mechanics approaches, respectively. The principal idea of the ALE approach is that an observer is neither located at a fixed position in space nor does it move with the material point, but can move arbitrarily\cite{isogeometric}\cite{ale}.
 	
 	\section {Equations} 
 	\label{sec:eqn}
 	The FSI problem consist of a fluid that is occupying a given domain $\Omega^F$  and a structure that occupies another domain $ \Omega^S $ which interact at the common boundary $ \tau $\cite{equation}.
 	
 	\subsection{Structural Equation}
 	The large displacement of the structure is governed by:
 	
 	\begin{equation}
 	\rho^S \frac{D^2 u}{D t^2} - \nabla .(F.S(u)) = \rho^S b^S \ in \ \Omega^S
 	\end{equation}
  
 	where u represents the displacements of the structure, $ b^S $ represents the body
 	forces applied on the structure. S represents the second Piola-Kirchhoff stress
 	tensor, $ \rho^S $ represents the density of the structure and F represents the deformation
 	gradient tensor.
 
 	\subsection{Fluid Flow Equation}	
 	The fluid equations to be solved are the incompressible Navier-Stokes equations	expressed in ALE formulation. The Navier Stokes equations can be derived from the conservation laws of mass and linear momentum and takes into consideration that the fluid is viscous. The equations in ALE formulation take the form:
 
    \begin{equation}
	\rho^F \frac{dv}{dt}|_\chi + \rho^F.c.\nabla v - 2 \ \mu \ \nabla .\varepsilon (v) + \nabla p  = \rho^F b^F \ in \ \Omega^F
	 \end{equation}
 	
 	\begin{equation}
	\nabla · v = 0 \ in \ \Omega^F
 	\end{equation}
 	
 	
 	Here v denotes the fluid velocity and p denotes the physical pressure. The fluid density and viscosity is given by $ \rho ^F $  and $ \mu $ respectively. The fluid body forces are	represented by $ b^F $ and $ \varepsilon (v) $ represents the strain rate tensor. As can be seen in (2) the ALE formulation comes into the equation in the fluid acceleration term and the convective term.
 	
 	\subsection{Coupling Equation}
 	At the interface $ \tau $, kinematic and dynamic continuity is required. The governing
 	kinematic coupling equation is:
 	
 	\begin{equation}
 	u_\tau (t) = d^F_\tau (t)
 	\end{equation}
 	
 	Here $d^F_\tau (t)$ represents the displacement of the fluid mesh nodes at the interface.
 	The dynamic coupling equation takes the form:
 		
 	\begin{equation}
 	h^S (t) + h^F (t) = 0
 	\end{equation}
 	
 	
 	where h = $\sigma$.n signifies the traction vector.
 	
 	\section {Boundary Conditions} 
 	\label{sec:bc} 
 	Regarding the boundaries in FSI, along solid-wall boundaries,the particle velocity is coupled to the rigid or flexible structure. The kinematic requirement is that no particle can cross the interface.  Due to the coupling between fluid and structure, one also needs to define some coupling conditions so the fluid and the structure do not overlap or detach during the motion\cite{equation}. These conditions depend on the fluid and also differ depending on whether the fluid is inviscid or viscous.
 	
 	
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